π, R { 2, 0, 3} , ( R),. R, [ 1, 1] = {x R 1 x 1} 1 0 1, [ 1, 1],, 1 0 1,, ( 1, 1) = {x R 1 < x < 1} [ 1, 1] 1 1, ( 1, 1), 1, 1, R A 1
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1 sup inf (ε-δ 4) ε-δ,,,, sup inf,,,,,, R R π( R) π 1
2 π, R { 2, 0, 3} , ( R),. R, [ 1, 1] = {x R 1 x 1} 1 0 1, [ 1, 1],, 1 0 1,, ( 1, 1) = {x R 1 < x < 1} [ 1, 1] 1 1, ( 1, 1), 1, 1, R A 1 sup A, inf A sup[ 1, 1] = sup( 1, 1) = 1, inf[ 1, 1] = inf( 1, 1) = 1 sup inf,, (, 1] = {x R x 1} (, 1) = {x R x < 1} , (, 1] (, 1) ,,, R,,,, R,, R (, 2 ) ( ). R 2
3 1.2 ( ). a > 0, b > 0 a < Nb N (Q R ). ε > 0, a a q < ε q 1.4. lim 1 n = 0, lim n = +, lim 1 2 n = 0 2 R A, i) M A ii) a A, a M M A, i) m A ii) a A, m a m A ii) M < x x A ii) ε > 0 M + ε A ii) ii) ε > 0 m ε A, i) A,, *1., A,, max A, min A *1 M M M M M, M M M M = M 3
4 A A = {a 1, a 2, a 3,, a n }, M = a 1, M < a k M = a k k = 2, 3,, n, M A, 2.1. R,,,,, A = (, 1] = {x R x 1} A = ( 1, 1] = {x R 1 < x 1} 1 1 A, m a < m a A *2 ii) m, A m, m > 1, 1 < a < m a A *3, m 1, m A A, [ 1, ) = {x R 1 x} [ 1, 1) = {x R 1 x < 1}, R ( 1, 1) = {x R 1 < x < 1} 2.1. A = [ 1, 0) (1, 2], 2.2. B = ( 2, 1] [ 1, 2), 2.3. C = {0}, 1, R D f(x), f(x) A A = {f(x) R x D} A R, D f A, A 3 R A, b R, a A a b, b A (upper bound), c R, a A c a *2 *3 4
5 , c A (lower bound) c A b A = ( 1, 1) 2 A, 1 A,, *4. R, A = ( 1, 1) A,, b A, x b x a A a b x, {x R x b} A,, R A A U(A), L(A) U(A) = {b R a b, a A}, L(A) = {c R c a, a A} A U(A) L(A) A A = ( 1, 1) U(A) = [1, ), L(A) = (, 1], A [ 1, 1] [ 1, 1) ( 1, 1] L(A) 1 A 1 U(A) 3.1. R A A M, M A, m, m A 3.2. R A A U(A), L(A) R A U(A) = A, L(A) = A U(A) A A = ( 1, 1), U(A) = [1, ) A A, A,, *4 5
6 c A b c A b A, U(A) L(A), A 4, R R A U(A) A, L(A) U(A) b U(A) b,, A a a b, A, A a a b (a ) b, A a a b (a ) b, b A, U(A) ( ) 4.1. A R, A A a a b (a ) b, A A a c a (a ) c, {a n } = {a 1, a 2, a 3, } R A = {a n R n = 1, 2, 3, }, A, {a n }, A, {a n } {a n } {a n R n = 1, 2, 3, } a n = ( 1) n {a n } { 1, 1, 1, 1, }, {a n R n = 1, 2, 3, } { 1, 1}, {a n } (a n ) = (a 1, a 2, a 3, ) 6
7 5 R A, U(A), L(A) A = U(A) = L(A) = R, U(A) L(A),, A, 5.1. A R, U(A) U(A). B = U(A), C U(A) R a A( ), a 1 < a, a 1 U(A) = B, a 1 C C C B = U(A), C A, A a a c C c c A, C c a > c A a a A b B, a b, c C, b B c < a b a A {b n }, {c n } : b 1 B, c 1 C, 2 d n = (b n + c n )/2 b n+1 = { dn (d n B) b n (d n C), c n+1 = { cn (d n B) d n (d n C) (, C c n d n b n B c n+1 = c n, b n+1 = d n, C c n d n b n B c n+1 = d n, b n+1 = b n b n+1 c n+1 = (b n c n )/2 ) c 1 c 2 c n b n b 2 b 1, {b n }, {c n }, *5., b n c n = b n 1 c n 1 2 * (ε-δ 2) 7
8 , b n c n = b 1 c 1 2 n 1 n 0, {b n }, {c n }, m = lim b n = lim c n, m U(A) = B i) m U(A) ii) x < m x U(A) i) A a, {b n } a b n (n = 1, 2, 3, ),, A a a lim b n = m, m A, m U(A) ii) x < m x, lim c n = m, ε = (m x)/2 > 0 c n m < ε c n c n m *6, c n x < c n m *7., c n C c n < a A a, x < a, x A, x U(A) i), ii) m U(A), 5.2. A R, L(A) L(A). A 1 A A = { a R a A} A c( L(A)), A a c a a c, c U(A ) U(A ), U(A ) m M = m M L(A) A a a A a m = M M a *6 c n b n, c n lim b n = m *7, c n A,, x < c n < m 8
9 , M L(A) M < x x, x < M = m, x < a a A, A a A a < x A a x A x L(A) M L(A), R A( ), U(A), m A (supremum), m = sup A, L(A), M A (infimum), M = inf A, 5.1, R A( ), A sup A( R) 5.4. R A( ), A inf A( R), A = { a R a A}, inf A = sup( A) A = ( 1, 1) U(A) = [1, ) sup A = 1, L(A) = (, 1] inf A = 1,, {a n } A = {a n R n = 1, 2, 3, }, lim a n = sup A. A = {a n R n = 1, 2, 3, },, 5.3, s = sup A R lim a n = s s a n A a n s, ε > 0 s ε(< s) *8, s ε < a A a, A a = a N N, a n, N n n a = a N a n *8 s 9
10 , s ε < a n s s a n < ε, ε > 0 N n = s a n < ε N, 5.6. {a n } lim a n = s = sup A A = {a n R n = 1, 2, 3, }, lim a n = inf A 5.1. A = [ 1, 1] [ 1 ] [ 2, 1 1 ] 2 3, 1 3 6,,,,,, * 9 1.( ) 2 a > 0, b > 0 na > b n 2.( ) 3.( ) R A *9 R (i), (ii), (iii) 2 A, B : (i) R = A B (ii) A B =, A, B (iii) a A, b B a b, : A, B A, B 10
11 4.( ) 2 a, b a < b a < q < b q 5.( ) {a n } sup{a n n = 1, 2, 3, } 6.( ) n = 1, 2, 3, I n = [a n, b n ], n 1 I n I n+1, lim (b n a n ) = 0 I n = {a}, lim a n = lim b n = a 7.( ), 1, 2, 3,,,, Q R,, * 10., R * 11, 2, 3, 1 2, { 1 2 3,, 3, 3 1 2,,, 1 2, Q R, Q, 3 Q 6.1 n = 1, 2, 3, R A n, A 1, A 2,, A k A 1 A 2 A k k, A n A n A n [ 1 ] n, 1 = {0} n ( 1 ) n, 1 = {0} n *10 Q R 0011 (ε-δ 3) *11, 11
12 n = 1, 2, 3, A n = [ 1 ] n, 1 n, A n * 12, A n = {0} ( 6.1). 6.1 ( ). I n = [a n, b n ] (n = 1, 2, 3,... ) I 1 I 2 I n I n+1, (1) (2) lim (b n a n ) = 0 I n I n = {a}, a = lim a n = lim b n. a 1 a 2 a n b n b 2 b 1, {a n }, {b n } a = lim a n, b = lim b n a, b n a n b n a b, [a, b] * 13, a = sup{a n n = 1, 2, 3, }, b = inf{b n n = 1, 2, 3, }, n a n a b b n, n = 1, 2, 3, [a, b] I n *12 A n *13 a = b [a, b] = {a} = 12
13 ,, [a, b] I n, a I n a. I n, (1) I = I n I, I c, n c [a, b] [a n, b n ], a n a c b b n, 0 c a b n a n lim (b n a n ) = 0 c a > 0 ε = b a b n a n = b n a n < ε = b a n c a = 0 a = c [a, b] I = {a}, a = b, [a, b] {a} (2) lim a n = lim b n = a 6.2 {a n },, {a 2n } = {a 2, a 4, a 6, } {a n+1 } = {a 2, a 3, a 4, }., k a nk n 1 < n 2 < n 3 <, n k = 2k, n 1 = 2 < n 2 = 4 < a 3 = 6 <,, {a n } {a nk } k a 2k {a 2k } = {a 2, a 4, a 6, } N = {1, 2, 3, } * 14 N( R) 1, m m, N K,, min K K 1 = K, n 1 = min K 1, i 2 K i = K i 1 {n i 1 }, n i = min K i, K n 1 < n 2 < n 3 < *
14 , K = {n k N k = 1, 2, 3, } {n k } = {n 1, n 2, n 3, }, N K, K = {n 1, n 2, n 3, } {a n } {a nk } K,,, a n = ( 1) n {a n }, {a 2n } = {1, 1, 1, }.,, {a n }, K = {k N n ( k) a n a k } (, {( 1) n } K = {2k k = 1, 2, 3, } = {2, 4, 6, }, {n} K =, {1/( 5 n)} K = {2} ) 2 (1) K (2) K,, (1) K, K = {n k k = 1, 2, 3, } {n k }, {a nk } a n1 a n2 a n3, {a nk } (2) K n 1 = 1, K, K, n 1 = (max K) + 1 n 1 K n 1 < n 2 a n1 a n2 n 2, n 2 K n 2 < n 3 a n2 a n3 n 3, n 4, n 5, {n k },, {n k } m m + 1 K K = {n k k = 1, 2, 3, }, 14
15 6.2, * ( ).., 6.2.,,, 6.3. n a n a n = 1 5 n {a n }, K = {k N n ( k) a n a k } K = {2}, 3 ( ) ( ) ( ) ( ) 2 I ( ) *15 Bolzano Weierstrass., 15
16 2.1 max A = 2, min A = B = [ 1, 1], max B = 1, min B = max C = min C = 0 ({0} ) [ 5.1 A = 1 ] 3, 1, sup A = 1 3 3, inf A = 1 3 [ 6.1 A = 1 ] n, 1 1/k 0 1/k, 0 A n., A, A a, a 0, 0 < 1/k < a k, 1/k < a (a > 0 ) a < 1/k (a < 0 ) a [ 1/k, 1/k] = A k A k a A a = 0, A = {0} 6.2. ( 6.1 ) 6.3 {a n }, a 1 a 2, a 1 < a 2 1 K n 3 a n < a n+1 < 0 < a 2 3 K, 2 K K = {2} 16
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